A one-dimensional line element model for transient free surface flow in porous media

A simple and physically sound model has been proposed to investigate the transient free surface flow behavior in porous media. Instead of the continuum-based model, the pore space between solid grains is conceptualized as a network of horizontal and vertical flow paths and then the flow paths are characterized by straight line elements. Based on the Darcy's law and flux equivalence between the continuum-based model and line element model, equivalent hydraulic parameters and continuity equation are derived through the control volume. By introducing a local coordinate system, unified governing equations and equivalent boundary conditions of the line element model are obtained in the form of one-dimensional formulations. The two-dimensional transient free surface flow problem is reduced to a one-dimensional problem of line element network and a finite-element algorithm is developed. The proposed one-dimensional line element model is verified by three illustrative examples, which indicate that the numerical results can satisfactorily match the analytical solutions and experimental observations from different sources. It is found that the solution accuracy has weak sensitivity of time-step size but sensitive to the spacing size, while numerical efficiency can be well guaranteed for different time-step sizes and spacing sizes. Based on the numerical results, the proposed model can give better performance for the evolution of free surfaces than the residual flow procedure, and is efficient to model transient surface flow with a complex drainage system.